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Theorem ressid 13203
Description: Behavior of trivial restriction. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressid  |-  ( W  e.  X  ->  ( Ws  B )  =  W )

Proof of Theorem ressid
StepHypRef Expression
1 ssid 3197 . 2  |-  B  C_  B
2 ressid.1 . . 3  |-  B  =  ( Base `  W
)
3 fvex 5539 . . 3  |-  ( Base `  W )  e.  _V
42, 3eqeltri 2353 . 2  |-  B  e. 
_V
5 eqid 2283 . . 3  |-  ( Ws  B )  =  ( Ws  B )
65, 2ressid2 13196 . 2  |-  ( ( B  C_  B  /\  W  e.  X  /\  B  e.  _V )  ->  ( Ws  B )  =  W )
71, 4, 6mp3an13 1268 1  |-  ( W  e.  X  ->  ( Ws  B )  =  W )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149
This theorem is referenced by:  submid  14428  subgid  14623  gaid2  14757  subrgid  15547  rlmsca  15952  rlmsca2  15953  pjff  16612  rlmbn  18778  ishl2  18787  evlrhm  19409  evl1sca  19413  evl1var  19415  pf1ind  19438  dchrptlem2  20504  lnmfg  27180  lmhmfgsplit  27184  pwslnmlem2  27195  dsmmfi  27204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ress 13155
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